Existence of a global weak solution for a system close to (REF )–(REF ) has been established in [1]}. The main difference relies in the definition of the boundary conditions () and () for the electrons. Moreover the result is obtained under restrictive assumptions on the parameters, the physical sense of which being unclear.
Let us highlight the misfit of model (REF )–(REF ) with respect to Onsager's reciprocal relation
[2]}, [3]} or its generalization beyond the linear setting  [4]}.
Since inertia is not intended to play a role in our model, one expects the fluxes to be proportional to the driving forces, which can be decomposed into chemical and electrical contributions:
\(J_i=-\sigma _i\partial _x \xi _i, \quad \mbox{ for }i = 1,2,\) 
